Necessary conditions for schatten class localization operators

We study time-frequency localization operators of the form A ϕ1,ϕ2 a, where a is the symbol of the operator and ϕ1, ϕ2 are the analysis and synthesis windows, respectively. It is shown in [3] that a sufficient condition for A ϕ1,ϕ2 a ∈ Sp(L2 (Rd)), the Schatten class of order p, is that a belongs to the modulation space Mp, ∞ (R2d) and the window functions to the modulation space M1. Here we prove a partial converse: if A ϕ1,ϕ2 a ∈ Sp(L2 (Rd)) for every pair of window functions ϕ1, ϕ2 ∈ S(R2d) with a uniform norm estimate, then the corresponding symbol a must belong to the modulation space Mp, ∞ (R2d). In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p = ∞ and p = 2, we recapture the results in [3], which were obtained by different methods